Metamath Proof Explorer


Theorem rgspnmin

Description: The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014)

Ref Expression
Hypotheses rgspnval.r
|- ( ph -> R e. Ring )
rgspnval.b
|- ( ph -> B = ( Base ` R ) )
rgspnval.ss
|- ( ph -> A C_ B )
rgspnval.n
|- ( ph -> N = ( RingSpan ` R ) )
rgspnval.sp
|- ( ph -> U = ( N ` A ) )
rgspnmin.sr
|- ( ph -> S e. ( SubRing ` R ) )
rgspnmin.ss
|- ( ph -> A C_ S )
Assertion rgspnmin
|- ( ph -> U C_ S )

Proof

Step Hyp Ref Expression
1 rgspnval.r
 |-  ( ph -> R e. Ring )
2 rgspnval.b
 |-  ( ph -> B = ( Base ` R ) )
3 rgspnval.ss
 |-  ( ph -> A C_ B )
4 rgspnval.n
 |-  ( ph -> N = ( RingSpan ` R ) )
5 rgspnval.sp
 |-  ( ph -> U = ( N ` A ) )
6 rgspnmin.sr
 |-  ( ph -> S e. ( SubRing ` R ) )
7 rgspnmin.ss
 |-  ( ph -> A C_ S )
8 1 2 3 4 5 rgspnval
 |-  ( ph -> U = |^| { t e. ( SubRing ` R ) | A C_ t } )
9 sseq2
 |-  ( t = S -> ( A C_ t <-> A C_ S ) )
10 9 elrab
 |-  ( S e. { t e. ( SubRing ` R ) | A C_ t } <-> ( S e. ( SubRing ` R ) /\ A C_ S ) )
11 6 7 10 sylanbrc
 |-  ( ph -> S e. { t e. ( SubRing ` R ) | A C_ t } )
12 intss1
 |-  ( S e. { t e. ( SubRing ` R ) | A C_ t } -> |^| { t e. ( SubRing ` R ) | A C_ t } C_ S )
13 11 12 syl
 |-  ( ph -> |^| { t e. ( SubRing ` R ) | A C_ t } C_ S )
14 8 13 eqsstrd
 |-  ( ph -> U C_ S )